A Partition Of Unity Finite Element Method For Wave Propagation Problem

The theory of elastic wave propagation can be applied to a variety of fields of science and technology, such as dynamic stress concentration in the structure, earthquake in seismology, oil exploration in geophysics, nondestructive evaluation of materials, strength and structure analysis in civil engineering, CT in medicine etc.The finite element method is one of the most effective numerical methods to simulate wave propagation problems. But there still have some unsolved problems. Indeed Zienkiewicz regarded the numerical simulation of the short wave problem as one of the two main unsolved problems of FEM. It is known that the choice of around 10 nodal points per wavelength is usually recommended to obtain an acceptable level of accuracy in order to solve the elastic wave equation. Consequently, this issue presents a severe limitation on the application of the FE procedure to the numerical simulation of the short wave problems in practice since the computational time needed is prohibitively expensive. In addition, it was reported that low-order finite elements exhibit poor dispersion properties, while higher-order finite elements raise some troublesome problems like the occurrence of spurious waves.Partition of unity finite element method (PUFEM), which was developed and studied in last decade, has the ability to include a priori knowledge about the local behavior of the solution in the finite element space. It enables us to construct finite element spaces that perform very well in cases where the classical finite element methods fail or are prohibitively expensive.In the dissertation, the mathematical foundation and recent work of PUFEM are reviewed firstly. Then, PUFEM was used to study the numerical simulation of elastic wave propagation as follows. (1) A PUFEM model is developed for the numerical simulation of transient elastic wave propagation. The finite element spaces are constructed by multiplying the standard finite element shape functions, which form a partition of unity, with the harmonic shape functions defined as the bases of the subspaces, which consist of a set of plane waves traveling in prescribed directions. (2) A special integration scheme, which is analytic for the elements with straight edges and semi-analytic for the elements with curved edges, for computing element matrices with the oscillatory nature of the integrands is developed. (3) The PUFEM model was used to simulate the wave propagation and scattering problem of anti-plane wave. The proper choice of the wave number k in the subspace of discretization approximation is discussed when the directions of wave propagation areknown. (4) A PUFEM model in the frame of generalized Biot u-U formulations proposed by Zienkiewicz and Shiomi is developed to simulate the problem of wave propagation in the saturated porous media when the fluid is compressible. (5) A program of PUFEM was developed to simulate the wave propagation in two dimensions in the framework of LAGAMINE, which is a general FEM code. The results of numerical examples exhibit that PUFEM is more efficient than FEM and the derived analytic integration scheme used for PUFEM saves a great deal of computational time as compared with standard Gauss-Legendre integration scheme. The present dissertation is outlined with the following chapters.In chapter 1, the numerical methods for simulating wave propagation are surveyed, such as finite difference method, pseudo-spectral method, finite element method, infinite element method, boundary element method, spectral element method and grid method etc. The PUFEM and its recent developments are discussed. Some existing work used to simulate wave propagation in the saturated porous media is also briefly reviewed. At the last of the chapter, the contents of the dissertation are outlined.In chapter 2, the theory and application of PUFEM are surveyed briefly. The concept of PUFEM is introduced after the discussion on the property of FEM approximation. The mathematical foundation of PUFEM and the relation of PUFEM to other numerical methods are presented. The applications of PUFEM to both Crack and the wave propagation problems are introduced in detail.In chapter 3, the PUFEM model for simulating transient elastic wave propagation is investigated. It is remarked that the existing work on PUFEM for simulations of wave propagations are only limited to steady-state scalar problems. The Shape function spaces of PUFEM for transient analysis are constructed, that makes the model able to directly handle the essential boundary conditions as it does in FEM. In each local shape function subspace, the wave direction and the wave number can be specified individually for a nodal point. The ill-conditioning problem of the effective stiffness matrix is also discussed in this chapter.In chapter 4, an analytic integration scheme for the element matrix in PUFEM is studied. The integrand of the matrix has highly oscillatory property; few existing integration schemes are successful in view of both accuracy and efficiency in computation. Here, a special integration scheme for computing the oscillatory function is developed. The integration scheme is analytic for the elements with straight edges and semi-analytic for the elements with curved edges. Therefore, the computational efficiency of the PUFEM is further enhanced as compared with the PUFEM using standard Gauss-Legendre integration scheme.In chapter 5, PUFEM is used to simulate the propagation and scattering problem of anti-plane wave, and the choice of the wave numbers k in the harmonic subspaces is suggested when the directions of wave propagation are known. When the directions of the wave propagation are known, the harmonic shape function subspaces of PUFEM can be enriched by taking different values of the wave number in the given few directions to obtain the numerical results superior to those obtained by the standard FEM in accuracy and efficiency, while the ill-conditioning of the effective stiffness matrix can be alleviated and even avoided. It is noted that the PUFEM can not only well simulate the propagation of a single harmonic wave but also simulate that of the complex wave composed of a number of harmonic waves.In chapter 6, the PUFEM is used to simulate the wave propagation problem in the saturated porous media. A PUFEM model in the frame of generalized Biot u-U formulations proposed by Zienkiewicz and Shiomi is developed to simulate the problem of wave propagation in the saturated porous media when the fluid is compressible. The PUFEM model is used to simulate the elastic wave propagation problems in drained and undrained saturated porous media.In chapter 7, the PUFEM program and data structures of the PUFEM code for simulating the wave propagation in two dimensions are described. The program was developed in the framework of the general FEM code LAGAMINE. Particularly, the functions of main subroutines, the flow charts of the PUFEM analysis, and the format of the data flow are explained and presented.The main contributions of the dissertation are summarized and the further work is suggested in the chapter 8.The work carried out in the dissertation is supported by National Key Basic Research and Development Program (973 Program, No. 2002CB412709) and the National Natural Science Foundation of China (No. 19832010).